Virasoro Frames and their Stabilizers for the E8 Lattice type Vertex Operator Algebra

The concept of a framed vertex operator algebra (FVOA) is new (cf. [DGH]). This article contributes to this theory with a full analysis of all Virasoro frame stabilizers in V , the important example of the E8 level 1 affine Kac-Moody VOA, which is isomorphic to the lattice VOA for the root lattice of E8(C). We analyze the frame stabilizers, both as abstract groups and as subgroups of Aut(V ) ∼= E8(C). Each frame stabilizer is a finite group, contained in the normalizer of a 2B-pure elementary abelian 2-group in Aut(V ), but is not usually a maximal finite subgroup of this normalizer. In particular, we prove that there are exactly five orbits for the action of Aut(V ) on the set of Virasoro frames, thus settling an open question about V in Section 5 of [DGH]. The results about the group structure of the frame stabilizers can be stated purely in terms of modular braided tensor categories, so this article contributes also to this theory. There are two main viewpoints in our analysis. The first is the theory of codes, lattices, markings and the resulting groups of automorphisms. The second is the theory of finite subgroups of Lie groups. We expect our methods to be applicable to the study of other FVOAs and their frame stabilizers. Appendices present aspects of the theory of automorphism groups of VOAs. In particular, there is a general result of independent interest, on embedding lattices into unimodular lattices so as to respect automorphism groups and definiteness. Department of Mathematics, University of Michigan, Ann Arbor, MI 48109-1109 USA. E-mail: rlgmath.lsa.umich.edu. Mathematisches Institut, Universität Freiburg, Eckerstraße 1, 79104 Germany. E-mail: [email protected]. The first author acknowledges financial support from the University of Michigan Department of Mathematics and NSA grant USDOD-MDA904-00-1-0011. 1991 Mathematics Subject Classification. Primary 17B69. Secondary 22E40, 20B25